1. The problem is to understand the matrix \(\begin{bmatrix}a & b \\ c & d\end{bmatrix}\). 2. This is a 2x2 matrix with elements \(a, b, c, d\) arranged in two rows and two columns. 3. Important properties include the determinant, which is calculated as \(ad - bc\). 4. The determinant helps determine if the matrix is invertible: if \(ad - bc \neq 0\), the matrix has an inverse. 5. The inverse of the matrix, if it exists, is given by \(\frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\). 6. This matrix can represent linear transformations in 2D space, such as rotations, scalings, and shears. 7. Understanding the matrix elements and operations on it is fundamental in linear algebra and its applications.